This statement in two chronicles is often repeated as fact in secondary sources, as yet another grievance Isabella of France had against her husband Edward II, along with supposedly 'removing' her children from her (rather than just setting up households for them in normal fashion), confiscating her lands and giving her a smaller income to live on (he did do that), and so on. Let's look at this logically. Why exactly would Edward II have wanted to annul his marriage to Isabella in 1325? What benefits could he have gained from it? Precisely none. He would, however, have suffered a whole pile of negative consequences. Edward was at war with Isabella's brother Charles IV of France from the autumn of 1323 (although military action did not begin until the summer of 1324) until a peace treaty was signed in June 1325. As Edward's biographer Seymour Phillips points out, "An attempted divorce in the conditions of 1325 would have been political madness, since it would have meant the repudiation of all agreements between England and France, which Edward and Isabella's marriage had been intended to strengthen, and would have plunged England into an immediate war with France" (shortly after peace had finally been established between the two countries).  Even during Edward's war with Isabella's brother, over Gascony, there is no reason to suppose that he considered annulment, or that it would have gained him anything. The only possible grounds Edward could have had for an annulment of his marriage in 1325 was consanguinity, as he and Isabella were second cousins once removed. They had been granted a papal dispensation for this, however. An annulment would have meant that their marriage had never been valid in the first place (as a marriage then could not simply be ended, in the way we understand divorce), which would have made Edward and Isabella's children illegitimate. Edward II spent much of 1325 negotiating marriages for three of their children with the royal houses of Spain. Why on earth would he have risked making them illegitimate? Why would he have acted in such fashion as to harm the position and status of his elder son and heir Edward of Windsor?
No proof of the two chroniclers' statement that Edward was trying to annul his marriage in 1325 has ever been discovered in the Vatican archives, nor is there any evidence that he ever wrote to the pope regarding this matter. He did send Thomas Dunheved to Pope John XXII in Avignon in 1325, it's true - but to complain about Alexander Bicknor, the archbishop of Dublin, whom Edward held responsible for his half-brother the earl of Kent's surrender at La Réole in September 1324, and who was not afraid to make his intense dislike of Hugh Despenser the Younger obvious and public. (Bicknor boasted that were he not a cleric, he would challenge Despenser to a duel.) John XXII, who made Dunheved a papal chaplain while he was visiting the papal court, wrote to Edward II in October 1325: "To the king, whose letters sent by Thomas Dunhevede, a Friar Preacher, the pope has received. The matter touching Alexander, archbishop of Dublin, cannot be heard in camera, but must be laid before the consistory...".  The Lanercost chronicler says that Edward also sent Robert Baldock to the pope regarding an annulment of his marriage. Baldock was chancellor of England from August 1323 until October 1326 , and did not leave the country, to my knowledge.
As for other chroniclers, the very well-informed author of the Vita Edwardi Secundi does not mention or even hint at an intended annulment. Neither does the royal clerk and chronicler Adam Murimuth, who knew the royal couple well and who visited the papal court in 1324. Neither does the author of the Flores Historiarum, who loathed Edward and who would have jumped on a chance to condemn him for humiliating his wife in this fashion. Neither does any other chronicler, even Jean Froissart, who a few decades later invented a tale of Isabella secretly fleeing to France from Winchelsea with her son in 1325, to escape from Edward's mistreatment of her (in fact she departed from Dover with a large retinue and, of course, Edward's full knowledge and consent). When Thomas Dunheved wrote to Edward II on 7 October 1325, he did not mention an annulment.  Edward was debating with his counsellors, by the end of 1324, the possibility of sending Queen Isabella to France to negotiate with her brother, and she departed for her homeland the following March. Why then would he suddenly decide to annul his marriage to her? It wasn't until late 1325 that it became apparent that the queen did not intend to return to England. Isabella herself wrote to her husband from France on 5 February 1326, addressing him as "our very dear and very sweet lord and friend" (nostre treschier et tresdouche seignur et amy) and informing him that although she wished nothing more than to return to his company and live and die with him, she did not dare, because of her fear of Hugh Despenser. Isabella did not mention that Edward was trying to annul their marriage; neither did any letters her brother Charles IV sent to Edward around this time or earlier (and Charles would have been utterly furious at this horrendous insult to his sister).  At Hugh Despenser's trial in November 1326, he was not accused of attempting to procure an annulment of Edward and Isabella's marriage. Why would Isabella not charge him with this, if he had done it? It was the gravest and most humiliating crime Despenser could have committed against her.
The Lanercost chronicler in his convent in the far north, although a useful source for events in Scotland and Scottish raids in the north of England, knew little of what had been going on at court a couple of decades earlier, while the Pauline annalist was merely reporting a rumour he had heard (ut vulgariter dicebatur), not stating it as definitive truth. Plenty of rumours were flying around England in 1325/26, including one reported in the Brut that Edward II intended to strangle his wife and his son Edward of Windsor to death. Edward was, apparently, informed of this after his deposition, and was - as any normal human being would be when accused of something as monstrous as wishing to murder his own child - deeply upset and horrified ("God knows, I thought it never, and now I would that I were dead! So would God that I were! For then were all my sorrow passed.")  Just because rumours existed does not automatically make them true, and it's a shame that writers continue to declare as fact that Edward was trying to divorce his wife without considering the ample evidence that he was doing no such thing, and without considering the logical implications and consequences of this act. It does fit so nicely into the popular Victim!Isabella school of thought, though, doesn't it? Precisely why Edward would have wanted to annul his marriage to the queen in 1325 and what he would have gained from it in exchange for taking such a huge risk is never actually explained; his supposed neglect of his queen and the nastiness of Hugh Despenser appear to be enough reason. Alison Weir claims in her biography of Isabella that "Lanercost's statement is to some extent corroborated by the fact that Dunheved was sent to the papal Curia on secret business at this time," but fails to notice the papal letter which demonstrates that Dunheved delivered Edward's letters regarding the archbishop of Dublin to the pope, and somehow confuses Thomas with his secular brother Stephen (as Lanercost clearly and correctly refers to the Dunheved brother in question as Thomas, I can't help but wonder how carefully she looked at the sources). Of course, it's entirely possible that Thomas Dunheved discussed other matters with the pope on the king's behalf, but we have no evidence, besides a rumour repeated in two chronicles written some years later, that an annulment of the king and queen's marriage was one of them.
I believe it is virtually certain that Edward II was not intending to annul his marriage in 1325; he had no reason to do so, and the consequences would have been disastrous for him, his children and his kingdom. Having said that, it is just possible that he was considering this course of action in the summer of 1326, by which time he knew that Isabella was going to betroth their son to the count of Hainault's daughter without his consent and invade his kingdom with an army. It's perhaps hardly surprising if he then decided that an annulment would solve his problems, and that even making his children and in particular his elder son and heir illegitimate would be worth it. I hasten to add there is no actual evidence that he was planning to ask the pope for an annulment and this is mere speculation: again, there are no documents in either the Vatican or England to confirm it. Edward met the bishop of Rochester, his ally Hamo Hethe, at Boxley Down in Kent in June 1326, and he and Hugh Despenser the Younger rode with the bishop back to Rochester. Edward asked Hethe if it were true that there had once been a queen who had disobeyed her husband and had therefore been deposed from her royal dignity. Hethe was having none of it, and retorted that whoever had told the king this had given him very bad advice.  This does sound as if perhaps Edward was then considering the possibility of annulment. This may be confirmed by two letters Edward sent to his son in 1326, then in France with Isabella (whether willingly or not). The first, written on 18 March, orders the thirteen-year-old not to marry without his father's consent, and to obey Edward "under pain of forfeiting all that he may to the king...". The second, written on 19 June, ends with the words "if the king find him contrary or disobedient hereafter to his will, by what counsel soever it may be, he will ordain in such wise that Edward [of Windsor] shall feel it all the days of his life, and that all other sons shall take example thereby of disobeying their lords and fathers." 
Do these letters, and Edward's remark to the bishop of Rochester, imply that Edward was now indeed considering an annulment of his marriage? I don't know, and if Edward did think along these lines, he took no action, and it's highly doubtful that the pope would have consented to annul his marriage anyway. If they do imply that a possible annulment was on the king's mind, it must be noted that this was months after Isabella had defied Edward and refused to return to him, and was planning an invasion of his kingdom. However justified Isabella may have thought her actions, seen from Edward's perspective it's hardly surprising that he was furious with her and (perhaps unfairly, given how young the boy was) with his son, even to the extent that he was willing to infuriate Isabella's brother Charles IV by annulling the marriage and willing to disinherit his son and thereby avert the threat to himself by making the boy illegitimate. If an annulment of Edward II and Isabella of France's marriage was ever on the cards, it came as a consequence of Isabella's actions against her husband, and was not a cause of them.
1) The Chronicle of Lanercost 1272-1346, ed. Herbert Maxwell, p. 249; Annales Paulini 1307-1340, in W. Stubbs, ed., Chronicles of the Reigns of Edward I and Edward II, volume 1, p. 337.
2) Seymour Phillips, Edward II, p. 483 note 169.
3) Calendar of Papal Letters 1305-1341, pp. 474, 479.
4) T.F. Tout, The Place of the Reign of Edward II in English History, p. 327.
5) F.D. Blackley, 'Isabella and the Bishop of Exeter', in T. A. Sandqvist and M. R. Powicke, eds., Essays in Medieval History Presented to Bertie Wilkinson, p. 226.
6) Isabella's letter is cited in Phillips, Edward II, p. 491. For Charles IV's correspondence with Edward II in the 1320s, see Pierre Chaplais, ed., The War of Saint-Sardos (1323-1325): Gascon Correspondence and Diplomatic Documents.
7) The Brut or the Chronicles of England, ed. F. W. D. Brie, volume 1, pp. 252-253.
8) Roy Martin Haines, 'Bishops and politics in the reign of Edward II: Hamo de Hethe, Henry Wharton, and the 'Historia Roffensis'', Journal of Ecclesiastical History, 44 (1993), pp. 605-606.
9) Calendar of Close Rolls 1323-1327, pp. 576-578.